2019 AMC 12A Problem 16

The numbers 1,2,\ldots,9 are randomly placed into the 9 squares of a 3 \times 3 grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?

\textbf{(A) }\frac{1}{21}\qquad\textbf{(B) }\frac{1}{14}\qquad\textbf{(C) }\frac{5}{63}\qquad\textbf{(D) }\frac{2}{21}\qquad\textbf{(E) } \frac17

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.